CONTENTS - Department of Mathematics and Statistics - University ...

X − μ
Example 1: If Z = , calculate X if Z = 1.96, μ = 8.5, and σ = 1.8
σ
• First make a “clean substitution”, i.e. substitute each of the known variables into the
equation without trying to simplify at all:
X − 8.5
1.96 =
1.8
• The division sign implies brackets around X – 8.5. We are “undoing” the equation, so
this part will be left to last.
( X − 8.5)
1.96 =
1.8
• This means we “undo” the value 1.8 first. Because the right hand side of the equation reads
“(X – 8.5) divided by 1.8”, we will multiply by 1.8, since multiplication is the inverse
operation to division:
1.96 × 1.8 = (X – 8.5)
• Because we added the brackets because of the original division sign and we have dealt with
the division, the brackets are no longer needed:
3.528 = X – 8.5
• To undo subtraction we perform the opposite operation, addition:
3.528 + 8.5 = X
• We have now rearranged the equation so that X is on its own.
X = 12.0 (one decimal place)
X − μ
Example 2: If Z = calculate X if Z = 2.58, μ = -2.5, σ = 0.85,
σ
n
and n = 60.
• Clean substitution: 2.58 =
X −−2.5
0.85
60
• Simplify a little:
2.58 =
X + 2.5
0.1097
• Solve the equation: 2.58 × 0.1097 = X + 2.5
0.2830 = X + 2.5
0.2830 − 2.5 = X
X = − 2.22 (to two d.p.)
Appendix 1 – Basic rules and concepts